Electron. Commun. Probab. 17 (2012), no. 31, 1–8.
DOI: 10.1214/ECP.v17-1921
ISSN: 1083-589X
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
On the infinite sums of deflated Gaussian products∗
Enkelejd Hashorva†
Lanpeng Ji‡
Zhongquan Tan§
Abstract
P∞
In this paper we derive the exact tail asymptotic behaviour of S∞ =
i=1 λi Xi Yi ,
where λi , i ≥ 1 are non-negative square summable deflators (weights) and Xi , Yi , i ≥
1, are independent standard
random variables. Further, we consider the
PGaussian
∞
p
tail asymptotics of S∞;p =
i=1 λi Xi |Yi | , p > 1, and also discuss the influence on
the asymptotic results when λi ’s are independent random variables.
Keywords: Gaussian products; infinite sums; random deflation; exact tail asymptotics; maxdomain of attraction; regular variation; chi-square distribution.
AMS MSC 2010: 60G70; 60G15.
Submitted to ECP on April 4, 2012, final version accepted on July 15, 2012.
1
Introduction and Main Result
Let Xi , Yi , i ≥ 1 be independent standard (zero mean and unit variance) Gaussian
random variables, and let λi , i ≥ 1 be non-negative constants. Define, for some fixed
integer n, the weighted sum
Sn =
n
X
λi Xi Yi .
i=1
Since a one-dimensional projection of a standard Gaussian random vector is distributed
as a Gaussian random variable, we have the stochastic representation
v
u n
uX
d
Sn = X1 t
λ2i Yi2 = X1 Zn ,
v
u n
uX
λ2i Yi2 ,
with Zn := t
i=1
(1.1)
i=1
d
where = stands for the equality of the distribution functions. Note in passing that (1.1)
holds for general random variables Y1 , . . . , Yn being independent of Gaussian random
variables X1 , . . . , Xn . Clearly, due to the symmetry about 0 of Xi , i ≥ 1, the following
stochastic representation
d
Sn =
n
X
λi Xi |Yi |
(1.2)
i=1
is also valid, and hence Sn is a symmetric (about 0) random variable. In this paper,
we will make use of the stochastic representations of Sn , following from the fact that
∗ Supported by the Swiss National Science Foundation Grant 200021-1401633/1.
† University of Lausanne, Switzerland. E-mail: enkelejd.hashorva@unil.ch
‡ Corresponding author at: University of Lausanne, Switzerland. E-mail: lanpeng.ji@unil.ch
§ Jiaxing University, China. E-mail: tzq728@163.com
On the infinite sums of deflated Gaussian products
Xi , i ≥ 1, are Gaussian. For notational simplicity, we consider in the following ordered
weights λi , i ≥ 1 meaning
λ := λ1 = λ2 = · · · = λm > λm+1 ≥ · · · ≥ 0.
In view of [7] (see also [11] p.168 and [16]), we have (set
P Zn2 > λ2 x
∼
n
Y
(1 − λ2i /λ2 )−1/2
i=m+1
Qn
m+1 (·)
=: 1 when m = n)
21−m/2 m/2−1
x
exp(−x/2), m ≤ n ≤ ∞(1.3)
Γ(m/2)
as x → ∞. The standard notation ∼ stands for asymptotic equivalence as the argument
tends to infinity, and Γ(·) denotes the Euler Gamma function.
In the light of Lemma 3.4 (presented in Section 3) we obtain
P {|Sn | > λx} ∼ P Zn2 > λ2 x exp(−x/2), n ≥ m,
(1.4)
which is shown in the special case λi = λ, i ≤ n in the first Lemma of [8].
An interesting quantity with application in statistics, insurance and several other
applied fields is the random variable
S∞ =
∞
X
λi Xi Yi ,
i=1
where the non-negative weights λi , i ≥ 1 are square summable, i.e.,
∞
X
λ2i < ∞.
(1.5)
i=1
The random variable S∞ appears as the distributional limits for various statistics; for
instance as shown by [9], S∞ is essential for the characterisation of continuous, separately exchangeable processes. Several examples of statistics given as infinite weighted
sums of Gaussian products appear naturally when dealing with U -statistics or row and
column exchangeable processes.
The main result of [8] gives an upper bound for the tail probability of the random
variable S∞ , namely
P {|S∞ | > x} ≤ KP {|Sm | > x} ,
x > 0,
(1.6)
Pm
with K some unknown constant and Sm = λ i=1 Xi Yi . Recall that m is the multiplicity
of the largest weight λ = λ1 , and by the square summability assumption on λi , i ≥ 1, m
is necessarily a finite integer.
The main goal of this contribution is to derive, instead of the bound above, the exact
tail asymptotic behaviour of |S∞ |. Our main result below gives such an asymptotic
expansion showing further connections with the tail asymptotic behaviours of Z∞ and
Sn for any n ≥ m.
Theorem 1.1. Let Xi , Yi , i ≥ 1 be independent standard Gaussian random variables.
For given constants λ := λ1 = λ2 = · · · = λm > λm+1 ≥ · · · ≥ 0 satisfying the square
summability criterion (1.5)
P {|S∞ | > λx} ∼
∞
h Y
(1 − λ2i /λ2 )−1/2
i=m+1
i 21−m/2
Γ(m/2)
ECP 17 (2012), paper 31.
xm/2−1 exp(−x)
(1.7)
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On the infinite sums of deflated Gaussian products
holds as x → ∞. Furthermore, as x → ∞, we have
P {|S∞ | > x} ∼
∞
h Y
i
(1 − λ2i /λ2 )−1/2 P {|Sn | > x} , for any n ≥ m,
(1.8)
i=n+1
and
2
P {|S∞ | > λx} ∼ P Z∞
> λ2 x exp(−x/2).
2
(1.9)
Extensions and Discussions
In this section we discuss two directions which provide natural extensions of the
main result presented in Theorem 1.1. Initially, motivated by the stochastic representation (1.2), we consider Sn;p , p > 0 defined by
Sn;p :=
n
X
p
λi Xi |Yi | .
i=1
The same reasoning as (1.1) yields
Sn;p
v
u n
uX
d
λ2i Yi2p =: X1 Zn;p .
= X1 t
i=1
We show in Lemma 3.1 below that both S∞;p and Z∞;p exist almost surely, and moreover
d
S∞;p = X1 Z∞;p . Since the tail asymptotic behaviour of Z∞;p is known (see [3], [10] and
[11]) applying Lemma 3.3 for any p > 1 we obtain
p
p 1
1
2
2m
1 1+p
− 1+p
− 1+p
2(1+p)
1+p
p
P {|S∞;p | > λx} ∼
p
x
exp − p
+p
x
,
(2.1)
2
π(1 + p)
where we use the same notation and assumptions as in Theorem 1.1. We note in passing
that for the cases when p ∈ (0, 1) the asymptotics can not be obtained similarly due to
the complexity of the tail asymptotic behaviour of Z∞;p (e.g., [10, 11, 12]).
Our second extension concerns the case of random deflators
Λ i = λ i Ri ,
i ≥ 1,
with Ri ∈ (0, 1], i ≥ 1 and λi ’s as above. A close inspection of (1.8) indicates that
P∞
the main contribution in the asymptotics of P {| i=1 Λi Xi Yi | > x} should be from the
Pm
quantity P {| i=1 Λi Xi Yi | > x}. However, the asymptotic behavior of the latter is, in
general, not easy to obtain, and thus in the following we discuss a special case that
R1 = · · · = Rm and lim
u→0
P {R1 > 1 − su}
= sγ , ∀s > 0,
P {R1 > 1 − u}
(2.2)
for some positive constant γ . The above assumption means that the function P {R1 > 1 − s}
is regularly varying at 0, so we have
P {R1 > 1 − s} = L(s)sγ ,
(2.3)
with L(·) a positive slowly varying function at 0, i.e., limu→0 L(us)/L(u) = 1, ∀s > 0. For
more details on the max-domains of attraction and regularly varying functions see [2],
[4] or [15].
ECP 17 (2012), paper 31.
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On the infinite sums of deflated Gaussian products
Theorem 2.1. Let Ri ∈ (0, 1], i ≥ 1, be independent random variables, which are further independent of Xi , Yi , i ≥ 1, such that (2.2) holds. Under the conditions of Theorem
1.1, we have
)
( ∞
X
Λi Xi Yi > λx ∼ KP {R1 > 1 − 1/x} xm/2−1 exp(−x),
P x → ∞,
i=1
with
∞
21−m/2 Γ(γ + 1) Y
K=
E
Γ(m/2)
i=m+1
(
λ2
1 − 2i Ri2
λ
−1/2 )
∈ (0, ∞).
Our final remark concerns the role of the deterministic weights λi , i ≥ m. In view of
our asymptotic results in the above theorems, the restriction that λi , i > m, are positive
is not necessary since only λ2i , i > m, appear. Therefore this condition can be replaced
by assuming instead
λ > |λi |
(2.4)
for all i > m.
3
Further Results and Proofs
We present first some lemmas and then continue with the proofs of Theorem 1.1 and
Theorem 2.1.
Lemma 3.1. Let λi , Xi , Yi , i ≥ 1, be as in Theorem 1.1. Then, for p > 0
P {|S∞;p | < ∞} = 1 = P {Z∞;p < ∞} ,
(3.1)
with Z∞;p being independent of X1 . Furthermore
d
S∞;p = X1 Z∞;p .
(3.2)
Proof. Since by (1.5)
2 E Z∞;p
=
∞
n
oX
E Y12p
λ2i < ∞,
i=1
it follows that P {Z∞;p < ∞} = 1. Similarly, using again (1.5)



v
u n

uX
E {|S∞;p |} ≤ lim inf E {|Sn;p |} = lim inf E |X1 |t
λ2i Yi2p
n→∞
n→∞


i=1
v
u n
∞
oX
u
≤ E {|X1 |} tE Y12p
λ2i < ∞
i=1
establishing thus (3.1). In order to show (3.2), it is sufficient to prove that the characteristic functions of both sides coincide. For any s ∈ R we have
2
s 2
E {exp(isX1 Z∞;p )} = E {E {exp(isX1 Z∞;p )} |Z∞;p } = E exp − Z∞;p
2
and
E {exp(isS∞;p )}
=



∞
∞

 Y
X
p
p
E exp is
λj Xj |Yj |  =
E {exp (isλj Xj |Yj | )}


j=1
=
∞
Y
j=1
2
s
E exp − λ2j |Yj |2p
2
j=1
ECP 17 (2012), paper 31.
2
s 2
= E exp − Z∞;p
2
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On the infinite sums of deflated Gaussian products
implying (3.2), and thus the proof is complete.
Lemma 3.2. Let ξi , i = 1, 2, be two non-negative independent random variables such
that, as x → ∞,
P {ξi > x} ∼ Ci xαi exp(−Li (x)xpi ), i = 1, 2,
(3.3)
with some positive constants Ci , pi , i = 1, 2, α1 , α2 ∈ R, and two positive measurable
functions Li (·), i = 1, 2. If limx→∞ Li (x) = Li > 0, i = 1, 2, hold, then
P {ξ1 ξ2 > x} ∼ P {ξ1∗ ξ2∗ > x} ,
x → ∞,
(3.4)
where ξ1∗ , ξ2∗ are two independent non-negative random variables such that for all x
large enough
P {ξi∗ > x} = Ci xαi exp(−Li (x)xpi ),
i = 1, 2.
Proof. The proof is similar to that of Lemma 3.2 in [5], and therefore omitted here.
Lemma 3.3. Under the assumptions of Lemma 3.2, if further
L1 (x) = L1 + o(x−p1 ),
x → ∞,
(3.5)
and L2 (x) = L2 ∈ (0, ∞) for all large x, then we have
P {ξ1 ξ2 > x}
∼
2πp L 1/2
2p2 α1 +2p1 α2 +p1 p2
2 2
2(p1 +p2 )
C1 C2 Ap2 /2+α2 −α1 x
p1 + p2
p1 p2
× exp(−(L1 A−p1 + L2 Ap2 )x p1 +p2 ),
x → ∞,
where A = [(p1 L1 )/(p2 L2 )]1/(p1 +p2 ) .
Proof. If L1 (x) = L1 > 0 for all x > 0, the claim is established by Lemma 2.1 in [1]. In
the light of Lemma 3.2, we can restrict our attention to the simpler case that
P {ξ1 > x} = C1 xα1 exp(−L1 (x)xp1 ) and P {ξ2 > x} = C2 xα2 exp(−L2 xp2 )
hold for x sufficiently large. As in the proof of Proposition 3.1 of [13], for some 0 < l1 <
p1
1 < l2 < ∞ (set zx = Ax p1 +p2 , A = [(p1 L1 )/(p2 L2 )]1/(p1 +p2 ) ), we have as x → ∞
p1
P {ξ1 ξ2 > x}
∼ C1 C2 p2 L2 xα1 zxp2 +α2 −1−α1
∼
C1 C2 p2 L2 xα1 zxp2 +α2 −α1
Z
Z
l2 Ax p1 +p2
p1
exp(−L1 (x/y)xp1 y −p1 − L2 y p2 ) dy
l1 Ax p1 +p2
l2
p1 p2
p2
exp(−x p1 +p2 [L1 (A−1 x p1 +p2 y −1 )(Ay)−p1 + L2 (Ay)p2 ]) dy.
l1
By (3.5) we can further write
P {ξ1 ξ2 > x}
∼
C1 C2 p2 L2 xα1 zxp2 +α2 −α1
Z
l2
p1 p2
exp(−x p1 +p2 [L1 (Ay)−p1 + L2 (Ay)p2 ]) dy.
l1
Since the function ψ(y) = L1 (Ay)−p1 + L2 (Ay)p2 attains its minimum in [l1 , l2 ] at 1,
applying the Laplace approximation we obtain
Z
l2
p1 p2
exp(−x p1 +p2 [L1 (Ay)−p1 + L2 (Ay)p2 ])dy
l1
√
p1 p2
2π
∼ q pp
exp(−ψ(1)x p1 +p2 ), x → ∞,
1 2
x p1 +p2 ψ 00 (1)
ECP 17 (2012), paper 31.
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On the infinite sums of deflated Gaussian products
where
p1
p2
ψ(1) = L1 [(p1 L1 )/(p2 L2 )]− p1 +p2 + L2 [(p1 L1 )/(p2 L2 )] p1 +p2 ,
and
ψ 0 (1) = 0,
ψ 00 (1) = L2 Ap2 p2 (p1 + p2 ) > 0,
hence the claim follows.
Lemma 3.4. Let Xi , Yi , i ≤ n, be independent standard Gaussian random variables. For
given weights λ = λ1 = λ2 = · · · = λm > λm+1 ≥ · · · ≥ 0, as x → ∞, we have
P {|Sn | > λx} ∼
n
Y
(1 − λ2i /λ2 )−1/2
i=m+1
21−m/2 m/2−1
x
exp(−x), n ≥ m.
Γ(m/2)
(3.6)
Proof.
By the representation
of Sn in (1.1) we need to derive the tail asymptotic of
P X12 (Zn2 /λ2 ) > x2 as x → ∞. The tail asymptotic of Zn2 implies that of X12 by taking
λ2 = · · · = λn = 0 and λ1 = 1. Hence (1.3) and Lemma 3.3 establish the claim.
The following result, which is restatement of Lemma 2.1 in [14], is crucial for the
proof of Theorem 1.1.
Lemma 3.5. Let G be a distribution function having an exponential tail with rate θ ≥ 0,
i.e.,
lim
x→∞
1 − G(x + y)
= exp(−θy), ∀y ∈ R,
1 − G(x)
and let H be another distribution function satisfying 1 − H(x) = o(1 − G(x)). If further,
R∞
MH (β) := −∞ eβx dH(x) < ∞ holds for some β > θ, then we have
1 − G ∗ H(x) ∼ MH (θ)(1 − G(x)),
x → ∞,
where G ∗ H denotes the convolution of distribution functions G and H .
Proof of Theorem 1.1 First proof: The result follows by (1.3), Lemma 3.1 and Lemma
3.3.
P∞
We present below an alternative proof. Write Sm,∞ =
i=m+1 λi Xi Yi , hence S∞ =
Sm + Sm,∞ . If λm+1 = 0, then the proof follows by (3.6). We consider therefore below
only the case λm+1 > 0. By the symmetry about 0 of S∞ and Sm for any x > 0 we have
P {|S∞ | > x} = 2P {S∞ > x} and P {|Sm | > x} = 2P {Sm > x} .
Pm
d
pPm
2
In view of (3.6), Sm = λ i=1 Xi Yi = λ
i=1 Xi Y1 is in the Gumbel max-domain of
attraction with constant auxiliary function a(x) = λ, i.e.,
P {Sm > x + yλ}
exp(−(x + yλ)/λ)
∼
= exp(−y),
P {Sm > x}
exp(−x/λ)
∀y ∈ R
as x → ∞. Since λm+1 ∈ (0, λ) (with the multiplicity denoted by m1 ) we have
P {Sm,∞ > x}
=
(1.3−1.4)
=
m+m
(
)
X1
(1.6) 1
1
P {|Sm,∞ | > x} ≤ KP λm+1 Xi Yi > x
2
2
i=m+1
( m
)!
X
o P λ
Xi Yi > x
= o(P {Sm > x}).
i=1
ECP 17 (2012), paper 31.
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On the infinite sums of deflated Gaussian products
Furthermore, in view of Lemma 3.1, for any s ∈ (1, λ/λm+1 ),
E {exp(sSm,∞ /λ)}

v
(
u X
2
2 !)
∞

X
u ∞
1
sλ
sλ
i
i
2
Yi  = E exp
Yi2
= E exp X1 t


λ
2
λ
i=m+1
i=m+1
(
!)
∞
∞
2
Y
Y
1 sλi
=
Yi2
E exp
=
(1 − s2 λ2i /λ2 )−1/2 < ∞.
2
λ
i=m+1
i=m+1



Consequently, applying Lemma 3.5 we obtain
P {Sm + Sm,∞ > xλ}
∼ E {exp(Sm,∞ /λ)} P {Sm /λ > x}
∞
X
λi Xi Yi P {Sm /λ > x}
= E exp
λ
i=m+1
" ∞
#
Y
2
2 −1/2
=
(1 − λi /λ )
P {Sm /λ > x} ,
i=m+1
hence the first claim follows from (3.6). Furthermore, Eq. (1.8) follows obviously from
(1.7) and (3.6). Finally, in view of (1.3) and (1.7), we establish (1.9), and thus the proof
is complete.
2
In the next lemma we present a result of Theorem 3.1 in [6] which will be used in
the proof of Theorem 2.1.
Lemma 3.6. Let ξ ∈ [0, 1] be a random variable with distribution function G such that
lim
1 − G(1 − x/u)
= xα , ∀x > 0
− G(1 − 1/u)
u→∞ 1
for some α ≥ 0. Assume that η is a positive random variable with distribution function
F in the Gumbel max-domain of attraction with some positive auxiliary function ω(·),
i.e.,
lim
u→∞
1 − F (u + x/ω(u))
= exp(−x), ∀x ∈ R.
1 − F (u)
If both ξ and η are independent, then
P {ξη > x} ∼ Γ(α + 1) 1 − G 1 −
1
uω(u)
(1 − F (x)),
x → ∞.
Proof of Theorem 2.1 We use the same idea as the proof of Theorem 1.1. Set next
Pn
i=1 Xi Yi for n ≥ 1. We only need to consider the case that λm+1 > 0 (with the
multiplicity denoted by m2 ≥ 1). With the aid of (1.4), (2.2) and Lemma 3.6, we obtain
that, as x → ∞,
Vn :=
P {R1 Vm > x − y}
P {R1 Vm > x}
=
∼
P {R1 |Vm | > x − y}
P {R1 |Vm | > x}
n
o
1
P R1 > 1 − x−y
P {|Vm | > x − y}
∼ exp(y),
1
P R1 > 1 − x P {|Vm | > x}
∀y ∈ R.
Furthermore, utilising (1.6) we have
(
P
∞
X
)
Λi Xi Yi > λx
i=m+1
)
(
)
m+m
∞
X
λi
λm+1 X 2
Xi Yi > x ≤ KP
Xi Yi > x
λ
λ i=m+1
i=m+1
(
≤P
ECP 17 (2012), paper 31.
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On the infinite sums of deflated Gaussian products
and from (1.4), (2.2) and Lemma 3.6
P {R1 Vm
Γ(γ + 1)
> x} ∼
P
2
1
R1 > 1 −
x
P {|Vm | > x}, x → ∞,
which, in the light of(1.4) and (2.3), implies
(
P
∞
X
)
Λi Xi Yi > λx
= o (P {R1 Vm > x}) ,
x → ∞.
i=m+1
Consequently, the claim follows by using Lemma 3.5 and Lemma 3.6.
2
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Acknowledgments. We would like to thank Prof. Mikhail Lifshits, Prof. Vladimir I.
Piterbarg, Prof. Oleg Seleznjev, Prof. Qihe Tang and the referees of the paper for
several suggestions which improved this contribution significantly.
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